Bioactive Taylor dispersion: moment generation theory

This thesis develops novel mathematical methods for the analysis of Taylor dispersion and active dispersion of swimming microorganisms. In the presence of gradients in the advecting velocity field, solutions of the governing advection–diffusion equation cannot generally be obtained in closed form. As a result, it is common to study moments of the longitudinal distribution to characterise dispersion. Many existing approaches rely on recursive calculations that become cumbersome at higher order or encounter computational difficulties when absorbing boundaries are present. Inspired by Aris’s method of moments, we develop a framework based on moment generating functions (MGFs) for analysing the longitudinal distribution of suspensions of tracers and swimming cells. Since the MGF encodes all moments of the distribution, this approach allows dispersion properties to be obtained more efficiently than recursive methods. Combined with perturbative spectral analysis, the framework yields general expressions for the drift and effective diffusivity—defined via the mean and variance of the distribution—that can be computed by solving a single eigenvalue problem on the cross-sectional domain. We present the general theory and compute solutions in simple examples, comparing results with the literature throughout. The method of MGFs is further extended using Floquet Theory to obtain an exact expression for the effective diffusivity of passive particles in oscillating shear flows. This provides an alternative to existing analyses of time-periodic flows, which typically rely on Fourier expansions or multi-scale techniques. We then introduce a complementary approach based on cumulant generating functions, demonstrating its efficiency for solving Taylor dispersion problems. Finally, this cumulant-based framework is used to develop a novel method for deriving swimming–advection–diffusion models describing the dispersion of swimming microorganisms. Under physical assumptions, we rigorously derive a model for dispersion of gyrotactic swimmers in a linear shear.